1. Field of the Invention
This invention relates generally to apparatus and methods of characterizing the optical response of fiber Bragg gratings.
2. Description of the Related Art
Fiber Bragg gratings (FBGs) have many applications in optical communications and optical fiber sensing. The effective refractive index profile Δn(z) of the fiber core mode (e.g., the LP01 mode, or a higher-order mode) as a function of the position z along the FBG generally varies roughly periodically with z, with an envelope that may vary along z. The effective refractive index Δn(z) determines most of the optical properties of the FBG, including but not limited to, the dispersion properties, the complex reflection impulse response hR(t), the complex transmission impulse response hT(t), the amplitudes |r(ω)|, |t(ω)| and the phases φr(ω), φt(ω) of the complex reflection spectrum r(ω) and the complex transmission spectrum t(ω), respectively, and the group delay in reflection
      ⅆ                  ϕ        r            ⁡              (        ω        )                  ⅆ          (      ω      )      and transmission
            ⅆ                        ϕ          t                ⁡                  (          ω          )                            ⅆ              (        ω        )              ,where ω is the angular frequency. These functions can also be expressed as functions of the optical wavenumber k, which has a simple relationship to the optical angular frequency ω, so it is a simple matter to switch between ω and k.
A first general method to determine Δn(z) of an FBG is to measure the complex reflection impulse response hR(t), which is the temporal dependence of the amplitude and phase of the signal reflected by the FBG when an extremely short optical signal is launched into the FBG. The complex reflection impulse response hR(t) can be measured directly by launching an ultra-short pulse (e.g., approximately 1 picosecond to approximately 30 picoseconds, depending on the grating length and period) into the FBG and measuring the temporal evolution of the reflected signal. This general method has the drawback of requiring that the width of the input laser pulse be much narrower than the impulse response of the FBG. In addition, interferometric techniques are used to measure both the phase and the amplitude of the complex reflection impulse response, and these techniques are complicated and inherently sensitive to noise or other fluctuations.
A second general technique to measure the complex reflection impulse response hR(t) is to use an interferometer to measure the wavelength dependence of both the amplitude and the phase of the optical signal reflected by the FBG (i.e., the complex reflection spectrum r(ω) or r(k)). The complex reflection spectrum r(ω) is the Fourier transform (FT) of the complex reflection impulse response hR(t), as described by A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE Journal of Quantum Electronics, Vol. 39, pp. 1018-1026, August 2003. The complex reflection impulse response hR(t) is then recovered from the complex reflection spectrum r(ω) by taking the inverse Fourier transform (IFT) of r(ω). As discussed below, the main difficulty of this general technique is that the measurement of the complex reflection spectrum is in general tedious, sensitive to noise, applicable to only special types of FBGs, and/or time-consuming.
The complex reflection spectrum r(ω)=|r(ω)|·exp(jφr(ω)) of an FBG is measurable using various interferometric measurement systems which are generally more complex and have stronger noise sensitivities than do measurement techniques which merely provide the amplitude of the reflection or transmission spectra. For example, in Michelson interferometry (e.g., as described by D. -W. Huang and C. -C. Yang, “Reconstruction of Fiber Grating Refractive-Index Profiles From Complex Bragg Reflection Spectra,” Applied Optics, 1999, Vol. 38, pp. 4494-4499), a tunable laser and an optical spectrum analyzer (OSA) are used to recover the phase of the complex reflection spectrum from three independent measurements.
In end-reflection interferometry (e.g., as described by J. Skaar, “Measuring the Group Delay of Fiber Bragg Gratings by Use of End-Reflection Interference,” Optics Letters, 1999, Vol. 24, pp. 1020-1022), the FBG is characterized using a tunable laser together with an OSA by measuring the spectral reflectivity that is caused by the interference between the FBG itself and the bare fiber end. This technique, however, is generally a destructive technique, since the bare fiber end must typically be only a few centimeters away from the FBG.
In low-coherence time reflectometry (e.g., as described by P. Lambelet et al., “Bragg Grating Characterization by Optical Low-Coherence Reflectometry,” IEEE Photonics Technology Letters, 1993, Vol. 5, pp. 565-567; U. Wiedmann et al, “A Generalized Approach to Optical Low-Coherence Reflectometry Inducing Spectral Filtering Effects,” J. of Lightwave Technol., 1998, Vol. 16, pp. 1343-1347; E. I. Petermann et al., “Characterization of Fiber Bragg Gratings by Use of Optical Coherence-Domain Reflectometry,” J. of Lightwave Technol., 1999, vol. 17, pp. 2371-2378; and S. D. Dyer et al., “Fast and Accurate Low-Coherence Interferometric Measurements of Fiber Bragg Grating Dispersion and Reflectance,” Optics Express, 1999, Vol. 5, pp. 262-266), a Michelson interferometer is illuminated with a broadband light source, and light reflected from the FBG, placed on one arm of the interferometer, and light reflected from a moveable mirror, placed on the reference arm of the interferometer, are coupled together and directed to a detector. This technique utilizes a slow mechanical scan to retrieve the impulse response of the FBG as a function of time, which makes this type of measurement time-consuming.
In low-coherence spectral interferometry (e.g., as described by S. Keren and M. Horowitz, “Interrogation of Fiber Gratings by Use of Low-Coherence Spectral Interferometry of Noiselike Pulses,” Optics Letters, 2001, Vol. 26, pp. 328-330; and S. Keren et al., “Measuring the Structure of Highly Reflecting Fiber Bragg Gratings,” IEEE Photon. Tech. Letters, 2003, Vol. 15, pp. 575-577), the slow scanning process is avoided by reflecting broadband laser pulses from the FBG and temporally combining these reflected pulses with their delayed replicas. This pulse sequence is then sent to an OSA, which records the power spectrum. The pulsed laser source of this technique has an autocorrelation function which is temporally much narrower (e.g., approaching a delta function) than the impulse response of the FBG. In other words, the recovery of the impulse response of a given FBG is limited in resolution to the autocorrelation trace of the pulsed laser source. Furthermore, the delay between the reflected pulse from the FBG and the input laser pulse has to be carefully adjusted to avoid overlap in the inverse Fourier transform domain, which makes the recovery impossible due to aliasing.
Typically, measurement systems which measure the amplitude of the reflection spectrum or of the transmission spectrum do not provide the missing phase information (i.e., φr(ω) and/or φt(ω)). The amplitude measurement, which is relatively simpler than the phase measurement, involves a tunable laser and an optical spectrum analyzer (OSA). Previously, various methods have been proposed to reconstruct the missing phase spectrum or group delay spectrum from only the amplitude measurement of |r(ω)| or |t(ω)|. The phase reconstruction technique presented by Muriel et al., “Phase Reconstruction From Reflectivity in Uniform Fiber Bragg Gratings,” Optics Letters, 1997, Vol. 22, pp. 93-95, only works for uniform gratings and has been independently shown to be unsuited for gratings with imperfections (J. Skaar and H. E. Engan, “Phase Reconstruction From Reflectivity in Fiber Bragg Gratings,” Optics Letters, 1999, Vol. 24, pp. 136-138). A similar technique has been suggested to improve the noise performance of the initial technique of Muriel et al., however this technique is still limited to only uniform gratings and the processing algorithm involves adjusting of filtering parameters, which depend on the FBG being characterized (K. B. Rochford and S. D. Dyer, “Reconstruction of Minimum-Phase Group Delay From Fibre Bragg Grating Transmittance/Reflectance Measurements,” Electronics Letters, 1999, Vol. 35, pp. 838-839).
One method of recovering the phase information from the amplitude data of FBGs was previously described by L. Poladian, “Group-Delay Reconstruction for Fiber Bragg Gratings in Reflection and Transmission,” Optics Letters, 1997, Vol. 22, pp. 1571-1573. The technique of Poladian utilized the fact that the transmission spectra of all FBGs belong to the family of minimum-phase functions (MPF) which have their phase and amplitude related by the complex Hilbert transform. In the technique of Poladian, using the Hilbert transformation, the phase or group delay of FBGs is recovered from only the measurement of the amplitude of the transmission spectrum |t(ω)|. This technique works very well but the numerical evaluation of the principle-value Cauchy integral in the Hilbert transform is not trivial and is rather noise-sensitive, as described by Muriel et al.